The hyperbolic behaviour of forced circle diffeomorphisms
Time: Tue 2026-09-08 10.00
Location: F3, Lindstedtvägen 26 & 28
Video link: https://kth-se.zoom.us/j/62550771976
Language: English
Subject area: Mathematics
Doctoral student: Kirthana Rajasekar , Analys, dynamik, geometri, PDE och talteori
Opponent: Professor Àlex Haro, University of Barcelona, Department of Mathematics and Computer Science
Supervisor: Docent Kristian Bjerklöv, Matematik (Avd.), Analys, dynamik, geometri, PDE och talteori
QC 2026-08-18
Abstract
The results in this thesis focus on the dynamical behaviour of forced circle diffeomorphisms, which can be modelled as skew-product maps on the two-dimensional torus. For notable forced systems such as chaotically forced Schrödinger cocycles and quasi-periodically forced Arnol’d circle maps, various numerical studies have revealed the occurrence of diverse dynamical behaviours. This thesis contributes to building a theoretical framework that reveals the conditions and mechanisms that govern the rich hyperbolic behaviour in certain forced systems.
In Paper A, we quantify the non-uniform hyperbolic behaviour of Schrödinger cocycles over expanding base maps, for an open class of potential functions. For cocycles with strongly expanding circle maps on the base, we establish asymptotic results for their Lyapunov exponents in the large coupling regime, which are uniform for all real values of energy. In Paper B, the focus is broadened to a class of circle maps forced by uniformly expanding circle endomorphisms. We establish open conditions, where the skew-product maps on the fibre are typically non-monotonic in the base variable, for which the Lyapunov exponents on the fibre are negative Lebesgue almost everywhere. This implies non-uniform hyperbolicity and consequently, local convergence of orbits on a fibre. Thus, the results in Paper A and Paper B contribute to the understanding of forced systems with highly chaotic forcing.
In Paper C, we study circle diffeomorphisms with two attracting and two repelling fixed points under quasi-periodic forcing. For a set of frequencies of positive measure, we prove the synchronisation of orbits on the same fibre. This proves the existence of a unique attracting and a unique repelling invariant graph. Thus, the results precisely describe the non-chaotic behaviour in these systems. Further, the geometric structure of these invariant graphs gives key insights on the non-uniform hyperbolicity of the system and statistical properties of Lebesgue almost every point on the two-dimensional torus.